Bilinearity rank of the cone of positive polynomials and related cones
Rudolf, Gabor and Noyan, Nilay and Papp, David and Alizadeh, Farid (2010) Bilinearity rank of the cone of positive polynomials and related cones. (Accepted/In Press) AbstractFor a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an ndimensional manifold C(K) in the space { (x, s)  x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when
optimizing over such cones, therefore it is natural to look for similar optimality conditions for nonsymmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several wellknown cones, in particular
the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or halfline there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential
polynomials. Item Type:  Article 

Uncontrolled Keywords:  Optimality conditions, positive polynomials, complementarity slackness, bilinearity rank, bilinear cones 

Subjects:  Q Science > Q Science (General) 

ID Code:  14714 

Deposited By:  Nilay Noyan 

Deposited On:  13 Oct 2010 10:17 

Last Modified:  20 Oct 2011 14:33 

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